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Abstract

An analogy between crowd synchrony and multi-layer neural network architectures is proposed. It indicates that many non-identical dynamical elements (oscillators) communicating indirectly via a few mediators (hubs) can synchronize when the number of delayed couplings to the hubs or the strength of the couplings is large enough. This phenomenon is modeled using a system of semiconductor lasers optically delay-coupled in either a fully connected or a diluted manner to a fixed number of non-identical central hub lasers. A universal phase transition to crowd synchrony with hysteresis is observed, where the time to achieve synchronization diverges near the critical coupling independent of the number of hubs.

Figures (5)

(a) Schematic of M non-identical elements interacting with each other via P hidden units (HUs), where σik stands for the coupling strength between element i and the kth HU. A dilution consists of setting a fraction of the couplings to vanishing values. (b) Schematic of an architecture with 2M elements and two non-identical HUs, with frequency detuning, wi between them, where only 2L out of the 2M elements have couplings to both HUs. Singly connected elements have coupling strengths σ1while elements coupled to two HUs have coupling strengths σ2.

Color chart of intensity correlation among all pair of lasers in the lower layer, ρ(i,j), for the architecture of Fig. 1(a) with M = 20 and P = 3 for three different coupling strengths. The lower layer lasers are sorted by increasing frequency. (a) σ = 24.55 where all pair correlations are below the threshold (below criticality). (b) σ = 24.67, correlation begins to form. (c) σ = 24.78, all pairs of lasers are correlated. The transition to crowd synchrony is identified at σc = 24.6398 as explained in the text.

(a) A power law behavior of synchronization time as a function of the deviation from σc. (b) Average correlation among all pairs of lower level lasers as a function of the normalized sigma (σ/σc) shows a hysteresis loop. (c) Decay of the correlation as a function of time, where σ is abruptly changed from a synchronized state, σ>σc, to σ = 0.995σc (see text for detail). Results indicate data collapse and are obtained for P = 3 and M in the range [20,100]. (d) Correlation decay exponent as a function of M obtained from the data of panel c.

(a) Critical coupling as a function of number of lower layer lasers, for different number of hidden units with the lack of dilution. (b) Critical coupling as a function of number of lower layer lasers, for different connectivity ratio, R, and with P = 4. The dashed lines are given by the middle expression of Eq. (4) with A~169.6.

Correlation among all lower layer lasers in Fig. 1(b) as a function of σ2, for different frequency detuning between the two HU lasers. As detuning is increased, the average correlation weakens. M = 40/80, L = 0.2M, σ1 = 1.3σc.